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CNA Seminar/Colloquium/Joint Pitt-CNA Colloquium
Julian Fischer
Max Planck Institute for Mathematics in the Sciences, Leipzig
Title: A higher-order large-scale regularity theory for random elliptic operators

Abstract: We develop a large-scale regularity theory of higher order for divergence-form elliptic equations with heterogeneous coefficient fields $a$ in the context of stochastic homogenization. Under the assumptions of stationarity and slightly quantified ergodicity of the ensemble, we derive a $C^{k,\alpha}$-"excess decay" estimate on large scales and a $C^{k,\alpha}$-Liouville principle for any $k\geq 2$: For a given $a$-harmonic function $u$ on a ball $B_R$, we show that its energy distance on some ball $B_r$ to the space of $a$-harmonic functions that grow at most like a polynomial of degree $k$ has the natural decay in the radius $r$, at least above some minimal (random) radius $r_0$. The Liouville principle states that the space of $a$-harmonic functions that grow at most like a polynomial of degree $k$ has (almost surely) the same dimension as in the constant-coefficient case. Our results rely on the existence of higher-order correctors for the homogenization problem, which we establish by an iterative construction.

Joint work with Felix Otto.

Date: Tuesday, May 3, 2016
Time: 1:30 pm
Location: Wean Hall 7218
Submitted by:  David Kinderlehrer